How to close a section of a sphere flat while retaining closest shape?
This can be accomplished pretty easily with the knife tool.



I first positioned the object how I wanted it, and then pressed k, clicked on one point outside the mesh, then moved the mouse to the opposed side, and clicked to create a cut like the one above. Then all you need to do is delete all the vertices of the sphere to the right of the cut, and the knife tool will have created a smooth edge for your sphere segment.

One possible issue is that the knife tool only does the half of the sphere facing you. In that case, delete the half of the sphere you can't see, and use a mirror modifier to duplicate the side with the knife cut so it makes a full sphere again.

Final result: